You can use git on BitBucket. As a matter of fact, I use BitBucket for my private projects. The benefit of using GitHub for public projects, on the other hand, is that it provides better visibility : more users, and possibly more contributors. It's the same kind of reason why people use facebook and not google+ (technical differences aside).

I'm not a git expert, but afaik it's easy to setup a project to be visible on both BitBucket and GitHub if need be.